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Circles Inscribed in Squares

Learn about the relationship between circles and squares when one is inscribed in the other, and explore the formulas that connect their properties.

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Circles Inscribed in Squares

Study Guide

Key Definition

When a $circle$ is inscribed in a $square$, the diameter of the $circle$ is equal to the side length of the $square$.

Important Notes

The perimeter of the square is equal to four times the diameter of the circle.
The area of the square is equal to the diameter of the circle squared.
The circumference of the circle is $\pi$ times the diameter.
The area of the circle is $\pi$ times the radius squared.

Mathematical Notation

$s$ denotes the side length of the square.

$d$ denotes the diameter of the circle.

$r$ denotes the radius of the circle.

Remember to use proper notation when solving problems.

Why It Works

The circle is inscribed in the square, so all sides of the square are tangent to the circle, creating right angles. This means the diameter of the circle is equal to the side length of the square.

Remember

The relationship between the side length of the square ($s$) and the radius of the circle ($r$) is $s = 2r$.

Quick Reference

Perimeter of the square:$4s$

Area of the square:$s^2$

Circumference of the circle:$2 \pi r$

Area of the circle:$\pi r^2$

Understanding Circles Inscribed in Squares

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

When a circle is inscribed in a square, the diameter of the circle is equal to the side length of the square.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

If a circle with radius $3 cm$ is inscribed in a square, what is the perimeter of the square?

Real-World Problem

Question Exercise

Intermediate

Garden Planning Scenario

You're planning a garden and want to create a square flower bed of side length 10 m with a circular path inscribed inside it (i.e., the circle touches all sides of the square). Calculate the area of the square flower bed and the area of the circular path (the region between the square and the circle).

Advanced Challenge

Thinking Exercise

Advanced

Think About This

Consider a square of side length $10 cm$. A circle is inscribed in this square. What is the area of the circle?

Recap

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Review key concepts and takeaways